Implementing Multiplayer
There is an addon for BGE made by agoose that has a multiplayer setup. I suggest you have a look at that and see if it provides an easy solution for you.



This is similar to the question "how do I carve a statue in wood - without using my hands". Multiplayer is not something you can implement in a few hours, or even a few days. A good multiplayer setup requires an understanding of a range of topics from advanced programming concepts through to the network stack and even some physics (and AI for more advanced lag compensation techniques).

So that this answer isn't completely useless as an answer, I will continue. 

Networking in python is done typically through the socket module, though other frameworks such as twisted are available. Because blender python scripts have to run inside a frame, typically ports have to be used in non-blocking mode, or a multithreaded system set up.

You also have to design the architecture of the system. Do you have a game instance that is the "true" state of the game and all the clients are just interfaces to it? Do you have purely client-side calculations and only send updates to what a player does? There are many more architectures around, have a read and take a pick.